power rule
d/dx [x^n] =
The power rule is a fundamental rule in calculus that allows us to find the derivative of a function raised to a power. It states that if we have a function f(x) = x^n, where n is a constant, then the derivative of f(x), denoted as f'(x) or dy/dx, is given by the formula:
f'(x) = n * x^(n-1)
This means that when we have a function that is a power of x, we can use the power rule to find its derivative. The power rule is especially useful when dealing with polynomials or functions involving exponents.
Let’s look at a couple of examples to illustrate how the power rule works:
Example 1:
Consider the function f(x) = x^3. To find its derivative, we can apply the power rule:
f'(x) = 3 * x^(3-1) = 3 * x^2
So, the derivative of f(x) = x^3 is f'(x) = 3x^2.
Example 2:
Now, let’s consider the function g(x) = 5x^4. We can use the power rule to find its derivative:
g'(x) = 4 * 5 * x^(4-1) = 20x^3
So, the derivative of g(x) = 5x^4 is g'(x) = 20x^3.
The power rule can be extended to more complex functions as well, by applying it repeatedly or combining it with other rules and properties of derivatives. It is a powerful tool that simplifies the process of finding derivatives of functions involving powers.
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